#1265277

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2126 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

(2,4)(5,6,9,12)(7,10,11,8)(21,22)(23,28)(24,29)(25,30)(26,27)(31,33)(32,36)(34,35);(5,7,9,11)(6,10,12,8)(14,17)(18,20)(22,27)(23,29)(30,35)(31,36);(5,6)(7,10)(8,11)(9,12)(13,14)(15,18)(16,17)(19,20)(21,23,26,29)(22,28,27,24)(25,31,34,36)(30,33,35,32);(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(21,24,26,28)(22,29,27,23)(25,33,34,32)(30,31,35,36);(5,8,9,10)(6,11,12,7)(13,15,16,19)(14,18,17,20)(21,25,26,34)(22,30,27,35)(23,31,29,36)(24,32,28,33)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=f,d^2=f,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1172473);